Junction conditions in quadratic gravity: thin shells and double layers

Transcription

1 Junction conditions in quadratic gravity: thin shells and double layers arxiv: v1 [gr-qc] 19 Oct 2015 Borja Reina, José M. M. Senovilla, and Raül Vera Física Teórica, Universidad del País Vasco, Apartado 644, Bilbao, Spain Abstract The junction conditions for the most general gravitational theory with a Lagrangian containing terms quadratic in the curvature are derived. We include the cases with a possible concentration of matter on the joining hypersurface termed as thin shells, domain walls or braneworlds in the literature as well as the proper matching conditions where only finite jumps of the energy-momentum tensor are allowed. In the latter case we prove that the matching conditions are more demanding than in General Relativity. In the former case, we show that generically the shells/domain walls are of a new kind because they possess, in addition to the standard energy-momentum tensor, a double layer energy-momentum contribution which actually induces an external energy flux vector and an external scalar pressure/tension on the shell. We prove that all these contributions are necessary to make the entire energy-momentum tensor divergence-free, and we present the field equations satisfied by these energy-momentum quantities. The consequences of all these results are briefly analyzed. PACS: Kd; d; w 1 Introduction Quadratic gravity refers to theories generalizing General Relativity GR by adding terms quadratic in the curvature to the Lagrangian density. The motivations for such modifications go back several decades ago see the critic paper [17], and today there is a general consensus that modern string theory see e.g. [1] and other approaches to quantum gravity see e.g. [16] present that structure, even with higher powers of the curvature tensor, in their effective actions. On the other hand, many times it is convenient to have a description of concentrated sources, that is, of concentrated matter and energy in gravity theories. These concentrated sources represent for instance thin shells of matter or braneworlds, or domain walls and impulsive matter or gravitational waves. They can mathematically be modelled by using 1

2 distributions, such as Dirac deltas or the like, hence, one has to resort to using tensor distributions. However, one cannot simply assume that the metric is a distribution because the products of distributions is not well defined in general, and therefore the curvature and Einstein tensor will not be defined. Thus, one must identify the class of metrics whose curvature is defined as a distribution, and such that the field equations make sense. For sources on thin shells, the appropriate class of metrics were identified in [11, 13, 22] in GR, further discussed in [9]. Essentially, these are the metrics which are smooth except on localized hypersurfaces where the metric is only continuous. We carry on a similar program in the most general quadratic theory of gravity, where extra care must be taken: the field equations, as well as the Lagrangian density, contain products of Riemann tensors, and, moreover, their second derivatives. Therefore, the singular distributional part such as the Dirac deltas cannot arise in the Riemann tensor itself, which can have at most finite jumps except in some very excepctional situations. We identify these and then concentrate on the generic, and more relevant, situation performing a detailed calculation using the rigorous calculus of tensor distributions see the Appendices for definitions and fundamental formulas with derivations to obtain the energy-momentum quantities on the shells. They depend on the extrinsic geometrical properties of the hypersurface supporting it, as well as on the possible discontinuities of the curvature and their derivatives. Surprisingly, and as already demonstrated in [18, 19, 20], a contribution of dipole type also appears in the energy-momentum content supported on the shell. This is what we call a double layer, in analogy with the terminology used in classical electrodynamics [12] for the case of electrodipole surface distributions. This analogy make the interpretation of these double layers somewhat misterious, as there are no negative masses ad thus no mass dipoles in gravitation. One of our purposes is to shed some light into this new mystery. From our results and those in [18, 19, 20], these double layers seem to arise when abrupt changes in the Einstein tensor occur. We also find the field equations obeyed by all these energy-momentum quantities, which generalize the traditional Israel equations [11], and describe the conservation of energy and momentum. Actually, we explicitly prove that the full energy-momentum tensor is divergence-free in the distributional sense by virtue of the mentioned field equations. Previous works on junction conditions in quadratic gravity include [2, 5, 7, 23] see also [6, 10] for the Gauss-Bonnet case, but none of them provided the correct full field equations with matter outside the shell, and they all missed the double-layer contributions, which are fundamental for the energy-momentum conservation. Maybe this is due to the extended use of Gaussian coordinates based on the thin shell: this prevents from making a mathematically sound analysis of the distributional part of the energy-momentum tensor, as the derivatives of the Dirac delta supported on the shell seem to be ill-defined in those coordinates. This is explained in detail in Appendix E. The paper is structured as follows. In Section 2 we present a purely geometric review on spacetimes with distributional curvature constructed by joining smooth spacetimes. The quadratic gravity field equations are introduced in Section 3, where the proper junction conditions for the description of thin shells layers are found. This is achieved by using distributional calculus, briefly reviewed in the Appendices. In Section 4, the matter content supported on the layer, i.e. the distributional part of the global energy momentum tensor, is found to contain a usual Dirac-delta term T µν δ together with another 2

3 contribution of double-layer type as mentioned above; the latter is denoted by t µν. Then, both T µν and t µν are computed in terms of geometrical quantities: the curvatures at either side of the layer and the extrinsic and intrinsic geometry of the hypersurface supporting it. The tensor T µν is decomposed into the proper energy momentum of the shell τ αβ, external flux momentum τ α and external pressure or tension τ corresponding to the completely tangent, tangent-normal and normal parts respectively. The double layer energy-momentum tensor distribution is found to resemble the energy-momentum content of a dipole surface charge distribution with strength µ αβ. This strength depends on the jump of the Einstein or equivalently the Ricci tensor at the layer. The allowed jumps of the curvature and its derivatives up to second order at the layer are determined in Section 5, again from a purely geometrical perspective. The general quadratic gravity field equations are obtained in Section 6. These are the inherited field equations on the layer, and they involve τ αβ, τ α, τ and µ αβ together with jumps on the layer of the spacetime energy-momentum tensor. These fundamental equations are the generalization of the Israel equations in GR to the general quadratic gravity theories. The covariant conservation of the full energy-momentum tensor with its distributional parts is explicitly demonstrated in Section 7, where we discuss how the double layer term is necessary for that. The field equations on the layer are analysed and further discussed in Section 8, where a classification of the junction conditions in the following cases are presented: proper matching, thin shells with no double layers, and pure double layers. In particular we find that if there is no double layer, then no external flux momentum τ α nor external tension τ can exist. Finally, in Section 9 some comparisons with the general GR case, and particular matchings of spacetimes, are provided. It is found that any GR solution containing a proper matching hypersurface will contain a double layer and/or a thin shell at the matching hypersurface if the true theory is quadratic. Therefore, if any quantum regimes require, excite or switch on quadratic terms in the Lagrangian density, then GR solutions modelling two regions with different matter contents will develop thin shells and double layers on their interfaces. In order to have a self-contained text, we devote some Appendices to review distributional calculus in manifolds and to present some useful general calculations with distributions. On the other hand, we present in Appendix E a we hope clarifying discussion about the difficulty, and in fact inconvenience, of using Gaussian coordinates for dealing with layers in quadratic Lagrangian theories, as it has been often done in the literature. 2 Junction: spacetimes with distributional curvature The space-time is given by an n + 1-dimensional Lorentzian manifold V, g. Let us consider the case where V, g possesses two different regions, say with different matter contents or different gravitational fields, separated by a border. This border will locally be a hypersurface V which can have any causal character, the physically more interesting case arising when it is timelike, which we will assume throughout in this paper. divides the manifold V into two regions V ±, as shown schematically in Fig.1. The metrics g µν ± are assumed to be smooth on V ± respectively µ, ν, = 0, 1,..., n. An important observation is that one can actually deal with two different coordinate systems on V ±. In fact, this is needed in most practical problems, as one is usually given two distinct solutions of the field equations that are to be matched: for instance, 3

4 V, g µν V +, g + µν n µ Figure 1: Schematic diagram of the situation under consideration: is a timelike hypersurface separating two regions of the space-time, V + and V, with corresponding smooth metrics g + and g. These two metrics also have well defined, definite limits, when approaching. If, and only if, the first fundamentals forms inherited by from V + and V agree, one can build a local coordinate system such that the entire metric is continuous across too. In that case, one can define a unique unit normal n µ, which we choose to point from V towards V +, as shown. one solution describing an interior with matter and another describing vacuum; or a background solution upon which a localized perturbation, such as a wave front or a shell of matter, propagates. Thus, we will be presented with two sets of local coordinates {x µ ±} with no relation whatsoever, each valid on the corresponding part V ± [11]. Two corresponding timelike hypersurfaces ± V ± which bound the regions V ± must be chosen on each ±-side to be matched. Of course, these two hypersurfaces are to be identified in the final glued spacetime, so that they must be diffeomorphic. The junction of V + with V by identifying + with depends crucially on the particular diffeomorphism used for this identification, hence we assume that this has already been chosen and is known. The glued global manifold V is defined as the disjoint union of V + and V with diffeomorphically related points of + and identified. This unique hypersurface is the matching hypersurface we denote simply by. Let {ξ a } be a set of local coordinates on a, b, = 1,..., n. Then, there are two parametric representations x µ ± = x µ ±ξ a of, one for each imbedding into each of V ±. As explained in the Appendix B, in order to have well defined curvature tensors in the sense of distributions we need a global metric which is at least continuous across. As is known [3, 15], this happens if and only if the two first fundamental forms h ± of inherited from both sides V ± agree. This agreement requires the equalities on h + ab = h ab, h± ab g± µνxξ xµ ± ξ a x ν ± ξ b 1 and implies that one can build local coordinate systems in which the metric can be 4

5 extended to be continuous across. The unique metric defined on the entire manifold that coincides with g ± in the respective V ± and is continuous across is denoted simply by g. Let n ± µ be the unit normals to as seen from V ± respectively. They are fixed up to a sign by the conditions n ± µ x µ ± ξ a = 0, n± µ n ±µ = 1 and one must choose one of them say n µ pointing outwards from V and the other n + µ pointing towards V +. Hence, the two bases on the tangent spaces at any point of {n +µ, xµ + ξ a } {n µ, xµ ξ a } agree and are then identified, so we drop the ± even though, in explicit calculations, one can still use both versions using the two coordinate systems on each side. We denote by e a the vector fields tangent to defined by the above imbeddings e a := xµ + ξ a = xµ ξ a. x µ + Note that { e a } are defined only on. The basis dual to {n µ, e µ a} is denoted by {n µ, ω a µ} where the one-forms ω a are characterized by x µ n µ ω a µ = 0, e µ b ωa µ = δ a b. The space-time version of the first fundamental form, which is now unique due to 1, is given by the projector to defined only on Notice that and that h µν = g µν n µ n ν. 2 n µ h µν = 0, h µρ h ρ ν = h µν, h µ µ = n, h µν e µ ae ν b = h ab e µ a = h ab ω b ν g νµ, e µ c ω c ν = h µ ν. Despite all the above, the extrinsic curvatures, or second fundamental forms, inherited by from both sides V ± will be, in principle, different, because the derivatives of the metric are not continuous in general. We denote them by K ± µν, and they are defined, as usual, by K ± µν := h ρ νh σ µ ± ρ n σ K ± µν = K ± νµ where only tangent derivatives are involved. Obviously n µ K µν ± = 0, thus only the nn + 1/2 components tangent to are non-identically vanishing. In terms of the imbeddings these components are given by K ± ab n± µ 2 x µ ± ξ a ξ + b Γ±µ ρσ 5 x ρ ± x σ ±, 3 ξ a ξ b

6 which is adapted to explicit calculations. These components correspond to the second fundamental form, defined as a tensor in by K ± ab = n µe ρ a ± ρ e µ b = eµ b eρ a ± ρ n µ. As shown in the Appendix C, the Riemann tensor can be computed in the distributional sense and acquires, in general, a singular part proportional to the distribution δ supported on which is defined in Appendix B : R α βµν = R +α βµνθ + R α βµν1 θ + δ H α βµν. 4 Here H α βµν is called the singular part of the Riemann tensor distribution and as shown in the Appendix C reads H α βλµ = n λ [ Γ α βµ ] nµ [ Γ α βλ ] where the square brackets always denote the jump of the enclosed object across according to the definition 137 given in Appendix B. We can provide a more interesting formula for this singular part. First, note that from the general formula for the discontinuities of derivatives 158 in Appendix D.2 we have [ α g µν ] = n α ζ µν for some symmetric tensor field ζ µν defined only on. This immediately gives which implies [ Γ α βλ ] = 1 2 ζα βn λ + ζ α λn β n α ζ βλ 5 H αβλµ = 1 2 n αζ βµ n λ + n α ζ βλ n µ n β ζ αλ n µ + n β ζ αµ n λ. 6 Note that this expression is invariant under the change ζ µν ζ µν + n µ X ν + n ν X µ for arbitrary X µ and thus only the part of ζ µν tangent to enters into the formula. Actually, one can prove the existence of C 1, piecewise C, changes of coordinates that remove any normal part of ζ µν arising in 5 see, e.g., [15]. Thus, from now on we assume that such a change has been performed and we will restrict ourselves to assuming that ζ µν is tangent to : n µ ζ µν = 0. But using 3 together with 5 we deduce K + ab K ab = n µ [ Γ µ ρσ ] e ρ a e σ b = 1 2 ζ ρσe ρ ae σ b 7 that is to say, the tangent part of ζ µν is characterized by the difference of the two ±- second fundamental forms. Thus, defining the jump on of the second fundamental form as usual [K µν ] := K + µν K µν, n µ [K µν ] = 0 8 we can rewrite 6 as the desired formula for the singular part of the Riemann tensor distribution: H αβλµ = n α [K βµ ] n λ + n α [K βλ ] n µ n β [K αλ ] n µ + n β [K αµ ] n λ. 9 This important formula informs us that the singular part of the Riemann tensor distribution vanishes if, and only if, the jump of the second fundamental form vanishes. By contractions on 4 we get with obvious notations: 6

7 The Ricci tensor distribution where its singular part is given by R βµ = R + βµ θ + R βµ 1 θ + H βµδ 10 H βµ := H ρ βρµ = [K βµ ] [K ρ ρ] n β n µ. 11 Thus, the singular part of the Ricci tensor distribution vanishes if, and only if, the jump of the second fundamental form vanishes, hence, if and only if that of the full Riemann tensor distribution does. The scalar curvature distribution whose singular part reads R = R + θ + R 1 θ + Hδ 12 H := H ρ ρ = 2 [K µ µ]. 13 It follows that the singular part of the scalar curvature distribution vanishes if, and only if, the jump of the trace of second fundamental form vanishes. And the Einstein tensor distribution with a singular part which is tangent to. G βµ := R βµ 1 2 g βµr = G + βµ θ + G βµ 1 θ + G βµδ 14 G βµ = [K βµ ] + h βµ [K ρ ρ], n µ G βµ = 0 15 A general result proven in [15] is that the second Bianchi identity holds in the distributional sense: ρ R α βµν + µ R α βνρ + ν R α βρµ = 0 from where one deduces by contraction β G βµ = 0 for the Einstein tensor distribution. By using 14 and the general formula 138 this implies 0 = β G βµ = n β [G βµ ] δ + β G βµ δ. 16 The second summand on the righthand side is computed according to the general formula 163 in Appendix D.1 β G βµ δ = g βρ ρ Gβµ δ = g βρ σ Gβµ n ρ n σ δ + g βρ h λ ρ λ G βµ δ = h ρλ λ G ρµ δ which, via 148 finally gives β G βµ δ = β G βµ K ρσg ρσ n µ δ. 7

8 Introducing this into 16 we arrive at 0 = δ n β [G βµ ] + β G βµ 1 2 n µg ρσ K + ρσ + K ρσ which implies, by taking the normal and tangent components, the following relations K + ρσ + K ρσg ρσ = 2n β n µ [G βµ ] = 2n β n µ [R βµ ] [R], 17 β G βµ = n ρ h σ µ [G ρσ ] = n ρ h σ µ [R ρσ ]. 18 These equations can also be obtained [11] by using part of the Gauss and Codazzi equations for on both sides, specifically 154 and 155 in Appendix D.1. Remark: A very important remark is that all formulae in this section are purely geometric, independent of any field equations, and therefore valid in any theory of gravity based on a Lorentzian manifold. 3 Quadratic gravity We are going to concentrate on the case of quadratic theories of gravity because, apart from its own intrinsic interest and as we are going to discuss, they allow for cases where gravitational double layers arise. Let us consider a quadratic theory of gravity in n + 1 dimensions described by the Lagrangian density L = 1 R 2Λ + a1 R 2 + a 2 R µν R µν + a 3 R αβµν R αβµν + L matter, 19 2κ where κ = c 4 /8πG is the gravitational coupling constant, Λ is the cosmological constant, a 1, a 2, a 3 are three constants selecting the particular theory, and L matter is the Lagrangian density describing the matter fields. Λ 1 and a 1, a 2, a 3 have physical units of L 2. The field equations derived from this Lagrangian read see e.g. [8] and references therein G αβ + Λg αβ + G 2 αβ = κt αβ, 20 where T αβ is the energy-momentum tensor of the matter fields derived from L matter, G αβ is the Einstein tensor and G 2 αβ encodes the part that comes from the quadratic terms: { G 2 αβ = 2 a 1 RR αβ 2a 3 R αµ R µ β + a 3R αρµν R ρµν β + a 2 + 2a 3 R αµβν R µν a a 2 + a 3 α β R a 2 + 2a 3 R αβ } 1 2 g αβ { a1 R 2 + a 2 R µν R µν + a 3 R ργµν R ργµν 4a 1 + a 2 R } 21 where := g µν µ ν is notation for the D Alembertian in V, g. If we want to find the proper junction conditions, or a description of thin shells or braneworlds in these theories, we have to resort to the distributional calculus see Appendices and use the formulas provided in the previous section. Then, in order to have the Lagrangian density as well as the tensor G 2 αβ well defined in a distributional sense so that the field equations 20 are sensible mathematically, one has to avoid any 8

9 multiplication of singular distributions such as δ δ. One could also hope for some cancellation of such terms between different parts of the Lagrangian, and of G 2 αβ, and this is discussed in the following subsection for completeness, but one has to bear in mind that these cancellations are probably ill defined anyway, and thus not relevant. In order to properly deal with products of distributions we would need a more general calculus, based e.g. on Colombeau algebras [4, 21], and hope that those cancellations certainly occur and are well defined. 3.1 Dubious possible cancellation of non-linear δ δ terms Let us start by examining the Lagrangian 19 recalling that the different curvature terms possess now singular parts proportional to δ, as given in 9 and its contractions 11 and 13. One could naively compute the products of these singular parts arising from the quadratic terms in 19 and collect them in a common-factor fashion. The result would be a term of type δ δ 2κ 1 [K ρ ρ] 2 + 2κ 2 [K αβ ][K αβ ] where we have introduced the abbreviations κ 1 := 2a 1 + a 2 /2, κ 2 := 2a 3 + a 2 /2. 22 to be used repeatedly in what follows. Then, one should require the vanishing of the term in brackets. A similar naive compilation should be performed with the non-linear distributions arising from the quadratic terms in the field equations 21. Imposing again that the full combination must vanish, and separating the resulting condition into its normal and tangent parts to we would find { κ1 [K ρ ρ] 2 + κ 2 3[K µν ][K µν ] 2[K ρ ρ] 2 } n α n β 23 +κ 1 [K ρ ρ]2[k αβ ] [K ρ ρ]h αβ + κ 2 2[K ρ ρ][k αβ ] [K µν ][K µν ]h αβ = The normal 23 and tangent 24 parts should vanish separately. In particular the trace of the tangent part reads κ 1 [K ρ ρ] 2 2 n + κ 2 2[K ρ ρ] 2 n[k µν ][K µν ] = We see directly that κ 1 = κ 2 = 0 solves 23 and 24, but in order to find all solutions we compute the determinant of the system 23 and 25. This yields 3 n[k ρ ρ] 2 [K ρ ρ] 2 [K µν ][K µν ] = Take first [K ρ ρ] = 0. Then, 23 and 25 reduce to κ 2 [K µν ][K µν ] = 0. If [K ρ ρ] 0 but [K ρ ρ] 2 = [K µν ][K µν ], 23 reads κ 1 + κ 2 [K ρ ρ] 2 = 0 and 25 is redundant since it becomes κ 1 + κ 2 [K ρ ρ] 2 2[K αβ ] [K ρ ρ]h αβ = 0. Thus, κ 1 + κ 2 = 0 would follow. Finally, if n = 3 and [K ρ ρ] 2 0, 23, 24 and 25 yield a new possibility not considered so far, summarized in [K αβ ] = 1 3 h αβ [K ρ ρ] = 1, [K αβ ][K αβ ] = 1 3, κ 1 κ 2 = In short, each of the following possibilities would seem to allow for the mutual annihilation of δ δ terms in 21 and in 19 : 9

10 1. κ 1 = κ 2 = [K ρ ρ] = 0 and κ 2 = [K ρ ρ] 2 = [K µν ][K µν ] = [K ρ ρ] 2 = [K µν ][K µν ] 0 and κ 1 + κ 2 = If the spacetime is 4-dimensional, κ 1 κ 2 = 0 and [K αβ ] = h αβ /3. Despite we have included this analysis here for completeness, we should not forget that these cases are not mathematically correct, and therefore they should not be taken seriously unless a more rigorous study is performed showing its feasibility. To understand the problems behind these naive calculations, we want to emphasize that there is no known way to give a sensible meaning to δ δ, let alone to things such as fδ δ. Thus, taking for granted that combinations of type f 1 δ δ + f 2 δ δ are related to f 1 + f 2 δ δ is, at least, dubious. Such difficulties were, for instance, noted in [6] for the Gauss-Bonnet case corresponding to the possibility 1 above, and one has to resort to analyzing thick shells, that is, layers with a finite width, or to a setting more general than distributions, such as the theory of nonlinear generalized functions described in [4, 21] and references therein. The thin shell formalism is simply not available. Therefore, we will abandon this route for now, and in this paper we will concentrate on the generic and well-defined cases analyzed in the next subsection. 3.2 Well defined possibilities: no δ δ terms The only mathematically well-defined possibilities in the available theory of distributions for the thin shell formalism, as just argued, are those where no δ δ term ever arises, leading to two different possibilities if we let aside the case of GR defined by a 1 = a 2 = a 3 = 0: 1. If either a 2 or a 3 is different from zero, then products of the Ricci tensor by itself, or by the Riemann tensor, appear in 21 and these are ill-defined if the singular parts 9 and 11 are non-zero. Thus, we must demand that the singular parts 9 and 11 vanish which happens, as proven above, if and only if the jump of the second fundamental form vanishes. Thus, in this situation it is indispensable to require [K µν ] = In this case, all the curvature tensors are tensor distributions associated to tensor fields see Appendix A with possible discontinuities across. Observe that then the Lagrangian density 19 is also a well defined, locally integrable, function. 2. If on the other hand a 2 = a 3 = 0, then only products of R by itself or by the Ricci tensor appear in 21, and thus it is enough to demand that R is a locally integrable function without singular part. Hence, in this case it is enough to require that 13 vanishes, that is to say, that the trace of the second fundamental form has no jump: [Kρ] ρ = 0. Observe that, again, the Lagrangian density 19 is in this case a well-defined locally integrable function. 10

11 In any of the above two possibilities, expression 20 with 21 has a remarkable property: there are no terms quadratic in derivatives of the curvature tensors. Taking into account that tensor distributions can be covariantly differentiated according to the rules explained in the appendices, the derivatives of the curvature tensors may have singular parts and still the field equations 20 are mathematically sound. This opens the door for the existence of matching hypersurfaces which represent double layers. Case 2 above was extensively treated in [18, 19, 20], where gravitational double layers were found for the first time. Therefore, we will here concentrate in the more general case 1, and thus we will assume hereafter that 28 holds. Notice that 28 coincide precisely with the matching conditions that are needed in General Relativity to avoid distributional matter contents, as follows from 15 together with the Einstein field equations. Once 28 is enforced, the lefthand side of the field equations 20 can be computed in the distributional sense. From 4 and 28 we know that the Riemann tensor distribution R αβµν = R + αβµν θ + R αβµν 1 θ, is actually associated to a locally integrable and piecewise differentiable tensor field. However, this tensor field may be discontinuous across, and thus [R αβµν ] may be nonvanishing. This leads, when computing covariant derivatives of R αβµν, to singular terms proportional δ and its derivatives. And these are going to arise in G 2 αβ. Thus, the energymomentum tensor on the righthand side of 20 must be treated as a tensor distribution and contain such terms, localized on, giving the energy-matter contents of the thin shell or double layer. In order to compute this matter content supported on we only have to calculate the singular part of G 2 αβ, because G αβ in 14 vanishes as follows from 28 with 15. But the only terms in 21 that are relevant for this singular part are α β R and R αβ and its contraction R. More precisely, we need to obtain the singular part of the expression 2a 1 + a 2 + 2a 3 α β R + a 2 + 4a 3 R αβ + 2a a 2 R g αβ = κ 1 + κ 2 α β R + 2κ 2 R αβ + κ 1 R g αβ. 29 This is the purpose of the next section. 4 Energy-momentum on the layer From 10 and the assumption 28 we know that R αβ = R + αβ θ + R αβ 1 θ from where, using the general formula 138 twice we deduce ν R αβ = ν R + αβ θ + νr αβ 1 θ + [R αβ]n ν δ, µ ν R αβ = µ ν R + αβ θ + µ ν R αβ 1 θ + [ νr αβ ]n µ δ + µ [Rαβ ]n ν δ.30 Via contractions here, or directly from 12, we also obtain R = R + θ + R 1 θ, ν R = ν R + θ + ν R 1 θ + [R]n ν δ, µ ν R = µ ν R + θ + µ ν R 1 θ + [ ν R]n µ δ + µ [R]nν δ 31 11

12 as well as R αβ = R + αβ θ + R αβ 1 θ + nρ [ ρ R αβ ]δ + g µν µ [Rαβ ]n ν δ, 32 R = R + θ + R 1 θ + n ρ [ ρ R]δ + g µν µ [R]nν δ. 33 Thus, we need to control the discontinuities of the Ricci tensor and the scalar curvature, and also to provide an expression for the singular distribution µ [Rαβ ]n ν δ supported on. The general formula 163 provides µ nν [R αβ ]δ = ρ [Rαβ ]n µ n ν n ρ δ + { h ρ µ ρ n ν [R αβ ] K ρ ρ [R αβ ]n µ n ν } δ. At this point we introduce a 4-covariant tensor distribution µναβ with support on, which takes care of the first summand here and is defined by or equivalently by µναβ := ρ [Rαβ ]n µ n ν n ρ δ µναβ, Y µναβ := [R αβ ]n ν n µ n ρ ρ Y µναβ dσ. Note that µναβ = νµαβ = µνβα. In summary, we have µ nν [R αβ ]δ = µναβ + { n ν h ρ µ ρ [R αβ ] + [R αβ ]K µν K ρ ρ n µ n ν } δ and therefore 30 becomes µ ν R αβ = µ ν R + αβ θ + µ ν R αβ 1 θ + µναβ + { [ ν R αβ ]n µ + n ν h ρ µ ρ [R αβ ] + [R αβ ]K µν K ρ ρ n µ n ν } δ. From the general formula 161, conveniently generalised, we have where [ ρ R βµ ] = n ρ r βµ + h σ ρ σ [R βµ ], 34 r βµ := n ρ [ ρ R βµ ], r βµ = r µβ 35 are the discontinuities of the normal derivatives of the Ricci tensor. Thus, we finally get µ ν R αβ = µ ν R + αβ θ + µ ν R αβ 1 θ + µναβ + { r αβ n ν n µ + n µ h ρ ν ρ [R αβ ] + n ν h ρ µ ρ [R αβ ] + [R αβ ]K µν K ρ ρ n µ n ν } δ. 36 Observe that the entire singular part is symmetric in αβ and in µν. From 36 we immediately get all the sought terms. First, by contracting with g αβ we find [18, 19, 20] µ ν R = µ ν R + θ + µ ν R 1 θ + µν + { bn ν n µ + n µ ν [R] + n ν µ [R] + [R]K µν K ρ ρ n µ n ν } δ 37 where [19, 20] b := r ρ ρ = n ρ ρ [R] 38 12

13 measures the discontinuity on the normal derivative of the scalar curvature, and [19] µν := g αβ µναβ is a 2-covariant symmetric tensor distribution with support on acting as follows 1 µν, Y µν := [R]n ν n µ n ρ ρ Y µν dσ; µν = ρ [R]nµ n ν n ρ δ. 39 Similarly, contracting 36 with g µν we readily get R αβ = R + αβ θ + R αβ 1 θ + r αβδ + g µν µναβ 40 where the last distribution acts as follows g µν µναβ, Y αβ = µναβ, g µν Y αβ = = [R αβ ]n ν n µ n ρ ρ Y αβ g µν dσ [R αβ ]n ρ ρ Y αβ dσ; g µν µναβ = ρ [Rαβ ]n ρ δ. Finally, by tracing either of 37 or 40 we easily derive R = R + θ + R 1 θ + b δ +, 41 where we have introduced the notation := g µν µν. Note that [18], Y = g µν µν, Y = [R]n ρ ρ Y dσ; = ρ [R]n ρ δ What we have proven is that the distribution G 2 αβ takes the following form where G 2 αβ = G2+ αβ θ + G2 αβ 1 θ + G αβ δ + G αβ 42 G αβ = 2κ 2 r αβ +κ 1 bg αβ κ 1 +κ 2 { bn α n β + n α β [R] + n β α [R] + [R]K αβ K ρ ρ n α n β }, 43 and after a trivial rearrangement G αβ = κ 1 gαβ αβ + κ2 2g µν µναβ αβ. 44 From 44 we define two new 2-covariant tensor distributions with support on [19]: Ω αβ := g αβ αβ = ρ [R]hαβ n ρ δ ; Ωαβ, Y αβ = [R]h αβ n ρ ρ Y αβ dσ 45 and Φ αβ := g µν µναβ 1 2 αβ 1 2 Ω αβ = ρ [Gαβ ]n ρ δ ; Φαβ, Y αβ = [G αβ ] n ρ ρ Y αβ dσ 46 1 There are some errata in the formulae for µν and Ω µν in [18], and for t µν in [19, 20]: in all cases Y must be replaced by Y µν. 13

14 recall that [G αβ ] is tangent to, n α [G αβ ] = 0, due to 17 and 18 together with the vanishing of G αβ as follows from 15 and 28. With these definitions, 44 is rewritten simply as G αβ = κ 1 +κ 2 Ω αβ +2κ 2 Φ αβ ; G αβ = ρ {κ1 + κ 2 [R]h αβ + 2κ 2 [G αβ ]} n ρ δ. 47 Given the structure 42, the field equations 20 can only be satisfied if the energymomentum tensor on the righthand side is a tensor distribution with the following terms T µν = T + µνθ + T µν1 θ + T µν δ + t µν 48 where T µν is a symmetric tensor field defined only on and t µν is by definition the singular part of T µν with support on not proportional to δ. We perform an orthogonal decomposition of T µν into tangent, normal-tangent and normal parts with respect to with T µν = τ µν + τ µ n ν + τ ν n µ + τn µ n ν 49 τ µν := h ρ µh σ ν T ρσ, τ µν = τ νµ, n µ τ µν = 0; τ µ := h ρ µ T ρν n ν, n µ τ µ = 0; τ := n µ n ν Tµν so that T µν = T + µνθ + T µν1 θ + τ µν + τ µ n ν + τ ν n µ + τn µ n ν δ + t µν. 50 Following [19, 20] the proposed names for the objects in 50 supported on, with their respective explicit expressions, are: 1. the energy-momentum tensor τ αβ on, given by κτ αβ = κ 1 + κ 2 [R]K αβ + κ 1 bh αβ + 2κ 2 r µν h µ αh ν β. 51 τ αβ is the only quantity usually defined in standard shells. 2. the external flux momentum τ α defined by κτ α = κ 1 + κ 2 α [R] + 2κ 2 r µν n µ h ν α. 52 This momentum vector describes normal-tangent components of T µν supported on. Nothing like that exists in GR. 3. the external pressure or tension τ κτ = κ 1 + κ 2 [R]K ρ ρ + κ 2 2r µν n µ n ν b, 53 Taking the trace of 51 one obtains a relation between b, τ and the trace of τ µν : κ τ ρ ρ + τ = κ 1 n + κ 2 b 54 The scalar τ measures the total normal pressure/tension supported on. Again, such a scalar does not exist in GR. 14

15 4. the double-layer energy-momentum tensor distribution t αβ, which is defined by κt αβ = G αβ = ρ {κ1 + κ 2 [R]h αβ + 2κ 2 [G αβ ]} n ρ δ 55 or, equivalently, by acting on any test tensor field Y αβ as κ t αβ, Y αβ = {κ 1 + κ 2 [R]h αβ + 2κ 2 [G αβ ]} n ρ ρ Y αβ. 56 t αβ is a symmetric tensor distribution of delta-prime type: it has support on but its product with objects intrinsic to is not defined unless their extensions off are known. As argued in [19, 20], t αβ resembles the energy-momentum content of double-layer surface charge distributions, or dipole distributions, with strength κµ αβ := κ 1 + κ 2 [R]h αβ + 2κ 2 [G αβ ], µ αβ = µ βα, n α µ αβ = We note in passing that κµ ρ ρ = κ 1 n + κ 2 [R], κt ρ ρ = κ 1 n + κ 2 58 The appearance of such double layers is remarkable, as massive dipoles do not exist. However, in quadratic theories of gravity they arise, as we have just shown, in the generic situation when thin shells are considered. In this case, t αβ seems to represent the idealization of abrupt changes, or jumps, in the curvature of the space-time. 5 Curvature discontinuities In the next section, we are going to derive the field equations satisfied by the energymomentum quantities 51, 52, 53 and 57 supported on. To that end, we have to perform a detailed calculation of the discontinuities of the field equations 20: they obviously include the discontinuities of the energy-momentum tensor T µν which must be related to the energy-momentum content concentrated on. The discontinuity of the lefthand side of 20 contains [G 2 αβ ] actually, we will only ] and this involves discontinuities of quadratic terms in the Riemann tensor, need n α [G 2 αβ such as [R 2 ], [R αβ R αβ ], [R αβµν R αβµν ], [RR αβ ], [R αµ R µ β ], [R αρµνr β ρµν ] and [R αµβν R µν ], as well as discontinuities of derivatives of the curvature tensors, such as [ α β R], [ R αβ ] or [ R]. Thus, we have to use systematically the rules 157 and either of 161 or 162 supplemented with 28, and we also need to have some knowledge on the discontinuities of the Riemann tensor and its derivatives. 5.1 Discontinuities of the curvature tensors Thus, let us start by controlling the allowed discontinuities of the Riemann tensor across. From the requirement 28 we know that ζ µν = 0 and thus [Γ α βµ] = 0. 15

16 Then, the general formula 158 gives [ ] λ Γ α βµ = nλ γ α βµ for some functions γ α βµ such that γ α βµ = γ α µβ, and therefore [R αβλµ ] = n λ γ αβµ n µ γ αβλ. 59 The antisymmetry of R αβλµ in [αβ] implies then that γ αβµ = B αβ n µ for some symmetric tensor B αβ = B βα defined only on. Hence γ αβµ = γ αβµ B αβ n µ, γ αβµ = γ βαµ and [R αβλµ ] = n λ γ αβµ n µ γ αβλ. However, the symmetry of γ αβµ in βµ implies γ α[βµ] B α[β n µ] = 0 as well as γ [αβµ] = 0 from where one easily derives and we recover the standard formula [15] γ αβµ = 2 γ µ[βα] = n α B βµ n β B αµ [R αβλµ ] = n α n λ B βµ n λ n β B αµ n µ n α B βλ + n µ n β B αλ. 60 As argued after formula 9, there is no loss of generality by assuming that B αβ is tangent to, i.e. such that B αβ n α = 0 = B αβ = B ab ω a αω b β. Given this plus the symmetry of B αβ there are n 2 + n/2 independent allowed discontinuities for the curvature tensor, all encoded in B ab. Successive contractions on 60 provide [R βµ ] = B βµ n βn µ [R], [R] = 2B ρ ρ, 61 or equivalently B βµ = [R βµ ] 1 2 [R]n βn µ = [G βµ ] h βµ[r]. 62 In other words, the nn + 1/2 allowed independent discontinuities of the Riemann tensor can be chosen to be the discontinuities of the -tangent part of the Einstein tensor or equivalently, of the Ricci tensor. 5.2 Discontinuities of the curvature tensors derivative Concerning the covariant derivative of the Riemann tensor, the general formula 161 leads to [ ρ R αβλµ ] = n ρ r αβλµ + h σ ρ σ [R αβλµ ], 63 where r αβµν is a tensor field defined only on and with the symmetries of a Riemann tensor. Using the second Bianchi identity for the Riemann tensor the previous formula implies n [ρ r αβ]λµ + h σ[ρ σ [R αβ]λµ ] = 0 16

17 which, on using 60 and after some calculations, implies the following structure for r αβµν : r αβµν = K αµ B νβ K αν B µβ + K βν B µα K βµ B να + µ B ρν ν B ρµ nα h ρ β n βh ρ α + α B ρβ β B ρα nµ h ρ ν n ν h ρ µ + n α n µ ρ βν n α n ν ρ βµ n β n µ ρ αν + n β n ν ρ αµ, 64 where ρ βµ is a new symmetric tensor field, defined only on and tangent to, n β ρ βµ = 0, which encodes the allowed new independent discontinuities of the covariant derivative of the Riemann tensor. There are nn + 1/2 of those again. As far as we know, relation 64 has only been derived in [15]. Contraction of 64 leads to the equation 34, but now with an explicit expression for the discontinuity of the normal derivative of the Ricci tensor which reads, on using 62 r βν = ρ βν + K ρ ρb βν [R]K βν K ρβ B ρ ν B ρβ K ρ ν n β ρ [G ρ ν] n ν ρ [G ρ β ] + n β n ν ρ α α, 65 where a natural orthogonal decomposition of r βµ appears: the first line is its complete tangent part which, given that ρ βν entails the allowed new independent discontinuities, is in itself a symmetric tensor field tangent to codifying those discontinuities. We are going to denote it by R βµ := h ρ β hσ µr ρσ = h ρ β hσ µn λ [ λ R ρσ ]; 66 the second line is its tangent-normal part, which is completely determined by the covariant derivative within of the discontinuity of the Einstein tensor n β h ν µr βν = ρ [G ρµ ]; 67 and finally, the third line gives the total normal component of r βµ, which can be related to the discontinuity 38 of the normal derivative of R by simply taking the trace r ρ ρ = b leading to r βµ n β n µ = b 2 + Kρσ [G ρσ ]. 68 Using this we get a useful relation for the trace of R αβ, that does not depend on ρ αβ 5.3 Second-order derivative discontinuities R α α = b 2 Kρσ [G ρσ ]. 69 Let us now consider the jumps in the second derivatives of the Ricci tensor. The starting point is equation 34. We can find an expression for the second summand there by differentiating 61 along and using the general rule 148 see Appendix D.1, h σ ρ σ [R βµ ] = 1 2 n βn µ ρ [R] + n µ Kβρ [R] 2B βλ K λ ρ + ρ B βµ. 70 The jumps of the second-order derivatives of the Ricci tensor, due to the general formula 161, can be written as [ λ ρ R βµ ] = n λ A ρβµ + h σ λ σ [ ρ R βµ ] 71 17

18 where A ρβµ = A ρβµ is a shorthand for A ρβµ = n λ [ λ ρ R βµ ]. The last term h σ λ σ[ ρ R βµ ] can be further expanded by first using 34 to obtain h σ λ σ [ ρ R βµ ] = K λρ r βµ + n ρ h σ λ σ r βµ + h σ λ σ h γ ρ γ [R βµ ]. and then computing the last summand here, which leads to h σ λ σ [ ρ R βµ ] = K λρ r βµ + 2n µ K βλ ρ [R] + [R]K ρβ K µλ 4K γ ρ λb γβ n µ +n β n µ 1 2 λ ρ [R] [R]K σ λ K ρσ + 2K γ ρ K σ λ B σγ + λ K γ ρ n ρ K σ λ K γ σ [R]hγβ 2B γβ nµ 1 2 n µn β n ρ K σ λ σ [R] + λ ρ B βµ 2K γ ρ B γβ K µλ n ρ K σ λ σ B βµ + n ρ h σ λ σ r βµ. 72 Let us stress the fact that all the terms in the first two lines in the above expression are symmetric in λρ. Concerning A ρβµ, let us first decompose it into normal and tangential parts by A ρβµ = n ρ A βµ + h γ ρa γβµ, A βµ := n ρ A ρβµ, A βµ = A µβ. In order to obtain an expression for h γ ρa γβµ we take the antisymmetric part of 71 with respect to [λρ], and contract with n λ. For the left hand side of 71 we use the Ricci identity applied to the Ricci tensor at both sides V ±, and take the difference of the limits on, so that [ λ ρ ρ λ R βµ ] = [R γ βρλr γµ ] + [R γ µρλr βγ ]. For the right hand side of 71, after the contraction with n λ, we get A ρβµ n λ n ρ A λβµ n λ h σ ρ σ [ λ R βµ ] = h γ ρa γβµ n λ h σ ρ σ [ λ R βµ ]. Isolating h γ ρa γβµ and using 72 for the last term of the above equation, it is then straightforward to obtain A ρβµ = n ρ A βµ + n ν [R γ βρνr γµ ] + n ν [R γ µρνr βγ ] + h σ ρ σ r βµ 1 2 n µn β K σ ρ σ [R] K σ ρ σ B βµ [R]K σ ρ K σβ n µ + 2K σ ρ K γ σb γβ n µ. 73 The expression for [ λ ρ R βµ ] now follows by combining 71 with 72 and 73. After little rearrangements, that reads [ λ ρ R βµ ] = n λ n ρ A βµ + n λ n ν [R γ βρνr γµ ] + [R γ µρνr βγ ] + 2n λ h ρ σ σ r βµ n µ n β n λ K ρ σ σ [R] 2n λ K ρ σ σ B βµ 2[R]n λ K ρ σ K σβ n µ +4n λ K ρ σ K γ σb γβ n µ + 2n µ K βλ ρ [R] + [R]K ρβ K µλ 4K γ ρ λ B γβ n µ 1 +n β n µ 2 λ ρ [R] [R]Kλ σ K ρσ + 2Kρ γ Kλ σ B σγ + K λρ r βµ + λ Kρ γ [R]hγβ 2B γβ nµ + λ ρ B βµ 2Kρ γ B γβ K µλ

19 We must stress the fact that there are still terms in 74, i.e. A βµ and r βµ, that are not completely independent. The contraction of 74 with g ρλ yields [ R βµ ] = A βµ + Kr βµ + n µ n β 1 2 [R] [R]Kρσ K ρσ + 2K σρ K γ ρ B σγ +2n µ h λ β ρ [R]K ρλ 2Kγρ ρ B γλ + 12 ρk ρλ ρk ργ B γλ +[R]K ρβ K ρ µ + B βµ 2K γ ρ K ρ µ B βγ, 75 while contracting with g βµ we obtain [18] [ ν µ R] = A ρ ρn ν n µ + 2n ν µ b 2n ν K λ µ λ [R] + bk νµ + ν µ [R]. 76 From any of the previous we readily have [ R] = A ρ ρ + bk + [R]. 77 The energy-momentum quantities will arise from the discontinuities of the normal components of the lefthand side of 20. In other words, we will only need to consider n α [G 2 αβ ]. Observe then that A βµ only appears in [ R βµ ], and since we only need the terms contracted with the normal once, in particular n β [ R βµ ], we are only interested in controlling n β A βµ. This can be done by using the identities 2 ρ R ρµ ± = µ R ± at both sides of, and taking the difference after one further differentiation: n ν [ ν ρ R ρµ ] = 1 2 nν [ ν µ R]. 78 The lefthand side here comes from 71 combined with 73 after one contraction, whereas for the righthand side we simply have to contract 76 with n ν. Equation 78 is thus found to be equivalent to n ρ A ρµ + n σ [RσR γ γµ ] + [R γµρσ R γρ ] + h βσ σ r βµ K βσ σ B βµ 1 n µ 2 [R]K ρσk ρσ K σβ KσB γ γβ = 1 A ρ 2 ρ n µ + µ b Kµ λ λ [R] Discontinuities of terms quadratic in the curvature Now, let us concern ourselves with the many terms in 21 quadratic in curvature tensors. To start with, using 61 with 157 we readily obtain [R αβ R αβ ] = 2[R αβ ]Rαβ = B αβ + 12 [R]nα n β Rαβ, 80 [RR αβ ] = R [R αβ ] + [R]Rαβ = R B αβ n αn β [R] + RRαβ, 81 [R 2 ] = 2[R]R. 82 Regarding n α [R αµ R µ β ], let us first consider the contraction nσ n µ [R γ σr γµ ]. The chain of equalities n σ n µ [R γ σr γµ ] = 2n σ n µ R γ σ[r γµ ] = [R]n γ n µ R γµ 83 19

20 follows from 61 and 157. Half-adding the two ± equations 154 and using the result in 83 we derive n σ n µ [R γ σr γµ ] = 1 2 [R]R R + K ρ ρ 2 K ρσ K ρσ. 84 Analogous procedures using the Gauss equation 149 accordingly yield n σ h µ ν[r γ σr γµ ] = B αν β K βα α K ρ ρ [R] αk α ν ν K ρ ρ, 85 n α n β [R αµβν R µν ] = 2R µν R µν + K ρ ρk µν K µσ K σ ν B µν, 86 n α h β λ [R αµβνr µν ] = β K λα λ K βα B αβ α K αβ β K ρ ρb βλ. 87 n α n β [R αρµν R β ρµν ] = 4n α n β [R αµβν R µν ] 4R ρσb ρσ, 88 n α h β λ [R αρµνr β ρµν ] = 2B αβ α K λβ λ K αβ, 89 [R αβµν R αβµν ] = 2n α n β [R αρµν R ρµν β ] = 8B αβ Rαβ R αβ + KρK ρ αβ K αρ K ρ β Discontinuities of the quadratic part [G 2 αβ ] We are now ready to compute the full n α [G 2 αβ ]. To keep track of the different terms, we split the compilation of terms in three parts, corresponding to the terms multiplied by either of the three constants a 1, a 2, a 3 in 21. Terms with a 1 : The terms in 21 that go with a 1 are G 2a 1 αβ := 2RR αβ 2 β α R 1 2 g αβr 2 + 2g αβ R, and we can compute their jump using 81, 76 and 77 to obtain and Terms with a 2 : n α n β [G 2a 1 αβ ] = 2[R]R αβn α n β + 2bK ρ ρ + 2 [R] 91 n α h β µ[g 2a 1 αβ ] = 2[R]R αβn α h β µ 2 µ b + 2K α µ α [R]. 92 The terms in 21 relative to a 2 are G 2a 2 αβ := 2R αµβν R µν β α R + R αβ 1 2 g αβ R µν R µν R. Before using 86 and 87 it is convenient to write down n α [ R αβ ] using 75 combined with 79, since some terms simplify. With the help of 86, 87, 76, 81 and it is then easy to get n α n β [G 2a 2 αβ ] = b 2 Kρ ρ + [R] [R]R R + K ρ ρ [R]K ρσk ρσ +R ρσ K ρσ + ρ µ [G ρµ ] + B µν Rµν R µν + KρK ρ µν, 93 n α h β µ[g 2a 2 αβ ] = 1 2 µb Kα µ α [R] + [R] α Kµ α 1 2 µk α R α µ + Kµ α ν [G να ] +B αβ β K αµ µ K αβ B αµ β K αβ K αβ β B αµ

21 Terms with a 3 : Regarding a 3 we have G 2a 3 αβ := 4R αµ R µ β +2R αρµνr ρµν β +4R αµβν R µν 2 β α R+4 R αβ 1 2 g αβr ργµν R ργµν. All terms have already appeared except for the last one, for which we use 90. Straightforward calculations lead to n α n β [G 2a 3 αβ ] = 4R αβ K αβ + 4 α β [G αβ ] + 4[G αρ ]K αβ K ρ β + 2 [R] +4B αβ Rαβ R αβ + K αβ Kρ ρ K αρ K ρ β, 95 n α h β µ[g 2a 3 αβ ] = +4K α µ β [G βα ] 4 α R α µ + 4K α µ α [R] 4 β B αµ K αβ Collecting all the above, we finally obtain n α n β [G 2 αβ ] = κ 1 +2[R] α K α µ 4B βµ α K αβ + 4B αβ β K αµ µ K αβ. 96 { bkρ ρ + [R] + 1 R R + K ρ 2 ρ 2 K ρσ K ρσ} +κ 2 { 2Rαβ K αβ + 2 α β [G αβ ] + 2B αβ R αβ R αβ + K αβ K ρ ρ K αρ K ρ β +2[G αµ ]K αβ K µ β + [R]} 97 n α h β µ[g 2 αβ ] = κ { 1 [R] α Kµ α µ Kρ ρ µ b + Kµ α α [R] } +κ 2 { 2 α R α µ + 2K α µ β [G βα ] + 2B αβ β K αµ µ K αβ + 2K α µ α [R] + [R] α K α µ 2B αµ β K αβ 2K αβ β B αµ }. 98 Remark: As a final remark, we would like to stress that all the discontinuities computed in this section 5 are purely geometrical, and therefore valid in any theory based on a Lorentzian manifold whenever 28 holds. 6 Field equations on the layer Relations 97 and 98 are the equations we were looking for, but we wish to rewrite them in terms of the derivatives of the energy-momentum quantities supported on given in and 57. Observe, first of all, that the three relations 66, 67 and 68 allow us to rewrite the energy-momentum contents supported on as follows κτ αβ = κ 1 + κ 2 [R]K αβ + κ 1 bh αβ + 2κ 2 R αβ, 99 κτ α = κ 1 + κ 2 α [R] 2κ 2 ρ [G ρα ], 100 κτ = κ 1 + κ 2 [R]K ρ ρ + 2κ 2 K ρσ [G ρσ ], 101 and using the definition of the double-layer strength 57 the last two here can be rewritten as τ α = ρ µ ρα, 102 τ = K ρσ µ ρσ

22 Now, a direct computation provides the following expressions for some combinations of derivatives of these objects: κ β τ αβ + Kρτ ρ α + α τ = κ 1 + κ 2 K β α β [R] + [R] β K αβ α Kρ ρ +κ 1 α b + 2κ 2 β R αβ + α K ρσ [G ρσ ] + K ρ ρ µ [G µα ], 104 κ τ αβ K αβ α τ α = κ1 + κ 2 [R] [R]K ρσ K ρσ 105 +κ 1 bk ρ ρ + 2κ 2 K ρσ R ρσ + α β [G αβ ]. 106 Using these, equations 98 and 97 become respectively after some rewriting using 150 and 151 and 20 κ n α h ρ β [T αρ] + α τ αβ + K ρ ρτ β + β τ = 2κ 2 { K αρ β [G αρ ] K ρ ρ α [G αβ ] + ρ [G αρ ]K αβ ρ [G αβ ]K αρ }, κ n α n β [T αβ ] + α τ α τ αβ K αβ = κ 1 + κ 2 [R] n α n β R αβ + K αβ K αβ Using now the definition of the strength 57 these become +2κ 2 [G µν ] n α n γ R αµγν + K ρ µk νρ. n α h ρ β [T αρ] + α τ αβ + K ρ ρτ β + β τ = K αρ β µ αρ K ρ ρ α µ αβ + ρ µ αρ K αβ ρ µ αβ K αρ n α n β [T αβ ] + α τ α τ αβ K αβ = µ µν n α n γ R αµγν + K ρ µk νρ. Recalling here the relations 102 and 103 between τ α and τ with the double-layer strength µ αβ, we finally obtain the following field equations n α h ρ β [T αρ] + α τ αβ = µ αρ β K αρ + ρ µ αρ K αβ ρ µ αβ K αρ, 107 n α n β [T αβ ] τ αβ K αβ = α β µ αβ + µ µν n α n γ Rαµγν + KµK ρ νρ. 108 These are the fundamental field equations satisfied by the energy-momentum quantities 51 and 57 within. They generalize the classical Israel equations of GR [11] and they are very satisfactory from a physical point of view. They possess an obvious structure with a clear interpretation as energy-momentum conservation relations. There are three type of terms in these relations. The first type is given by the corresponding first summands on the lefthand side. They simply describe the jump of the normal components of the energy-momentum tensor across. Therefore, they are somehow the main source for the energy-momentum contents in. The second type of terms are those on the lefthand side involving τ αβ, the energy-momentum tensor in the shell/layer. We want to remark that the first equation 107 provides the divergence of τ αβ. Finally, the third type of terms are those on the righthand side, involving the strength µ αβ of a double layer. These terms act also as sources of the energy-momentum contents within, combined with extrinsic geometric properties of and curvature components in the space-time. An alternative version of 107, after use of the Codazzi equation 152, reads n α h ρ β [T αρ] + α τ αβ = µ αρ n σ R σαλρh λ β + K αβ ρ µ αρ ρ µ αβ K αρ. 109 Note that the allowed jumps in the Riemann tensor 60 lead to n σ [R σαλρ ]h α γ h λ β hρ ξ = 0 and therefore the term µ αρ n σ Rσαλρ hλ β in the last formula can be written simply as µ αρ n σ R σαλρ h λ β. 22

23 7 Energy-momentum conservation The divergence of the lefthand side of the field equations 20 vanishes identically due to the Ricci and Bianchi identities, and therefore, the conservation equation for the energymomentum tensor µ T µν = 0 follows. In our situation, however, we are dealing with tensor distributions, and with 20 considered in a distributional sense. The question arises if whether or not the energy-momentum tensor distribution 50 is covariantly conserved. We know that the Bianchi and Ricci identities hold for distributions see Appendices, hence it is expected that the divergence of the T µν vanishes when distributions are considered. In this section we prove that this is the case, when taking into account the fundamental field equations 107 and 108. The following calculation can be alternatively seen, therefore, as an independent derivation of 107 and 108 from the covariant conservation of T µν. From 48 and 138 we directly get α T αβ = n α [T αβ ]δ + α T αβ δ + α t αβ. 110 Let us first compute the middle term on the righthand side. From the orthogonal decomposition 49 α T αβ δ = α {τ β + τn β } n α δ + α {τ αβ + τ α n β } δ and using the general formula 163 the second summand can be expanded to get α T αβ δ = α {τ β + τn β } n α δ + α τ αβ τ αρ K αρ n β + τ α K αβ + n β α τ α δ so that with the help of 102 we get α T αβ δ = α {τ β + τn β } n α δ + α τ αβ τ αρ K αρ n β K αβ ρ µ ρα n β α ρ µ αρ δ. 111 With respect to the last term in 110, on using definitions 56 and 57 we can write for any test vector field Y β and using the Ricci identity α t αβ, Y β = t αβ, α Y β = µ αβ n ρ ρ α Y β dσ = µαβ n { ρ α ρ Y β + R β σρ α Y σ} dσ = µ αβ n ρ α ρ Y β dσ n ρ µ ασ Rραβσδ, Y β. The first integral here can be expanded as { µ αβ n ρ α ρ Y β dσ = µ αβ α n ρ ρ Y β K αρ ρ Y β} dσ = n ρ ρ Y β µ ασ K ασ n β α µ αβ dσ µ αβ K αρ ρ Y β + n σ Y σ β K ρ dσ = τn β + τ β n ρ ρ Y β dσ + Y β ρ µ αβ K αρ n β µ ασ K αρ K σ ρ dσ = α {τ β + τn β } n α δ, Y β + ρ µ αβ K αρ n β µ ασ K αρ K ρ σ δ, Y β 23